Abstract
We present an extensive numerical study of spectral statistics and eigenfunctions of quantized triangular billiards. We compute two million consecutive eigenvalues for six representative cases of triangular billiards, three with generic angles with irrational ratios with $\ensuremath{\pi}$, whose classical dynamics is presumably mixing, and three with exactly one angle rational with $\ensuremath{\pi}$, which are presumably only weakly mixing or even nonergodic in case of right triangles. We find excellent agreement of short- and long-range spectral statistics with the Gaussian orthogonal ensemble of random matrix theory for the most irrational generic triangle, while the other cases show small but significant deviations which are attributed either to a scarring or superscarring mechanism. This result, which extends the quantum chaos conjecture to systems with dynamical mixing in the absence of hard (Lyapunov) chaos, has been corroborated by analyzing distributions of phase-space localization measures of eigenstates and inspecting the structure of characteristic typical and atypical eigenfunctions.
Highlights
Classical ergodic theory provides a fairly satisfactory classification of statistical properties of classical dynamical systems
We have presented an extensive numerical study of quantum chaos in two classes of triangle billiards
The first class (A) consist of generic triangles where all angles have irrational ratios with π and in the second class (B) exactly one of the angles has a rational ratio with π
Summary
Classical ergodic theory provides a fairly satisfactory classification of statistical properties of classical dynamical systems. It has been demonstrated numerically [12] that such triangular billiards are ergodic and mixing Their classical trajectories belong to two-dimensional surfaces of finite genus defined by the angles [19] and cannot be ergodic on three-dimensional energy surfaces Their quantum spectral properties belong to neither the chaotic nor integrable universality classes, but have potentially less universal intermediate spectral statistics. The most recent study of spectral statistics in generic triangular quantum billiards [24] considered spectra of up to 1.5 × 105 levels in a family of class A billiards, where in most cases intermediatelevel statistics were observed. Using several short- and long-range measures of spectral statistics, such as level spacings, spacing ratios, number variance, spectral form factor, and mode fluctuations, we confirm that the class A triangle billiards conform to GOE universality.
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